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Properties

Label 2-72-9.7-c3-0-5
Degree 22
Conductor 7272
Sign −0.253+0.967i-0.253 + 0.967i
Analytic cond. 4.248134.24813
Root an. cond. 2.061102.06110
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.78 + 4.38i)3-s + (−8.65 − 14.9i)5-s + (1.18 − 2.05i)7-s + (−11.5 − 24.4i)9-s + (26.1 − 45.2i)11-s + (−6.84 − 11.8i)13-s + (89.9 + 3.70i)15-s − 82.9·17-s − 126.·19-s + (5.72 + 10.9i)21-s + (27.1 + 47.0i)23-s + (−87.4 + 151. i)25-s + (139. + 17.2i)27-s + (106. − 184. i)29-s + (112. + 194. i)31-s + ⋯
L(s)  = 1  + (−0.535 + 0.844i)3-s + (−0.774 − 1.34i)5-s + (0.0640 − 0.110i)7-s + (−0.427 − 0.904i)9-s + (0.715 − 1.23i)11-s + (−0.146 − 0.253i)13-s + (1.54 + 0.0636i)15-s − 1.18·17-s − 1.53·19-s + (0.0594 + 0.113i)21-s + (0.246 + 0.426i)23-s + (−0.699 + 1.21i)25-s + (0.992 + 0.123i)27-s + (0.681 − 1.18i)29-s + (0.649 + 1.12i)31-s + ⋯

Functional equation

Λ(s)=(72s/2 ΓC(s) L(s)=((−0.253+0.967i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.253 + 0.967i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(72s/2 ΓC(s+3/2) L(s)=((−0.253+0.967i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.253 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 7272    =    23⋅322^{3} \cdot 3^{2}
Sign: −0.253+0.967i-0.253 + 0.967i
Analytic conductor: 4.248134.24813
Root analytic conductor: 2.061102.06110
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ72(25,⋅)\chi_{72} (25, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 72, ( :3/2), −0.253+0.967i)(2,\ 72,\ (\ :3/2),\ -0.253 + 0.967i)

Particular Values

L(2)L(2) ≈\approx 0.416486−0.539974i0.416486 - 0.539974i
L(12)L(\frac12) ≈\approx 0.416486−0.539974i0.416486 - 0.539974i
L(52)L(\frac{5}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
3 1+(2.78−4.38i)T 1 + (2.78 - 4.38i)T
good5 1+(8.65+14.9i)T+(−62.5+108.i)T2 1 + (8.65 + 14.9i)T + (-62.5 + 108. i)T^{2}
7 1+(−1.18+2.05i)T+(−171.5−297.i)T2 1 + (-1.18 + 2.05i)T + (-171.5 - 297. i)T^{2}
11 1+(−26.1+45.2i)T+(−665.5−1.15e3i)T2 1 + (-26.1 + 45.2i)T + (-665.5 - 1.15e3i)T^{2}
13 1+(6.84+11.8i)T+(−1.09e3+1.90e3i)T2 1 + (6.84 + 11.8i)T + (-1.09e3 + 1.90e3i)T^{2}
17 1+82.9T+4.91e3T2 1 + 82.9T + 4.91e3T^{2}
19 1+126.T+6.85e3T2 1 + 126.T + 6.85e3T^{2}
23 1+(−27.1−47.0i)T+(−6.08e3+1.05e4i)T2 1 + (-27.1 - 47.0i)T + (-6.08e3 + 1.05e4i)T^{2}
29 1+(−106.+184.i)T+(−1.21e4−2.11e4i)T2 1 + (-106. + 184. i)T + (-1.21e4 - 2.11e4i)T^{2}
31 1+(−112.−194.i)T+(−1.48e4+2.57e4i)T2 1 + (-112. - 194. i)T + (-1.48e4 + 2.57e4i)T^{2}
37 1+32.2T+5.06e4T2 1 + 32.2T + 5.06e4T^{2}
41 1+(250.+433.i)T+(−3.44e4+5.96e4i)T2 1 + (250. + 433. i)T + (-3.44e4 + 5.96e4i)T^{2}
43 1+(−6.41+11.1i)T+(−3.97e4−6.88e4i)T2 1 + (-6.41 + 11.1i)T + (-3.97e4 - 6.88e4i)T^{2}
47 1+(104.−181.i)T+(−5.19e4−8.99e4i)T2 1 + (104. - 181. i)T + (-5.19e4 - 8.99e4i)T^{2}
53 1−371.T+1.48e5T2 1 - 371.T + 1.48e5T^{2}
59 1+(2.90+5.03i)T+(−1.02e5+1.77e5i)T2 1 + (2.90 + 5.03i)T + (-1.02e5 + 1.77e5i)T^{2}
61 1+(−302.+523.i)T+(−1.13e5−1.96e5i)T2 1 + (-302. + 523. i)T + (-1.13e5 - 1.96e5i)T^{2}
67 1+(377.+653.i)T+(−1.50e5+2.60e5i)T2 1 + (377. + 653. i)T + (-1.50e5 + 2.60e5i)T^{2}
71 1−43.4T+3.57e5T2 1 - 43.4T + 3.57e5T^{2}
73 1−671.T+3.89e5T2 1 - 671.T + 3.89e5T^{2}
79 1+(−324.+562.i)T+(−2.46e5−4.26e5i)T2 1 + (-324. + 562. i)T + (-2.46e5 - 4.26e5i)T^{2}
83 1+(67.2−116.i)T+(−2.85e5−4.95e5i)T2 1 + (67.2 - 116. i)T + (-2.85e5 - 4.95e5i)T^{2}
89 1+206.T+7.04e5T2 1 + 206.T + 7.04e5T^{2}
97 1+(−726.+1.25e3i)T+(−4.56e5−7.90e5i)T2 1 + (-726. + 1.25e3i)T + (-4.56e5 - 7.90e5i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−13.79046279110824336740199264041, −12.48547093477995770349339206081, −11.59919085558911877592863510127, −10.62252492073193778467606160344, −9.010583880172123994242527988560, −8.434815063939118023077311059299, −6.33911369973064737489732563184, −4.87175120988279087593451480516, −3.87815165367553907896411203179, −0.46735021420415633227617212473, 2.31182166758298249459667846443, 4.40554722920985747657343631650, 6.60528768023941667007322682854, 6.97884433557927590712820022617, 8.432314732670337572744040575712, 10.31317045437170488974332898503, 11.30129440937769803144658808139, 12.09353166230473628395622834386, 13.24622707243852200909950482399, 14.66475365449898672695810052762

Graph of the ZZ-function along the critical line